# Questions tagged [fundamental-solution]

Questions on fundamental solutions of an ordinary differential equation.

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### Solving the fundamental solution of the origin of $\Delta u=\Delta_x u+\frac{a}{r}u_r+u_{rr}=0$

The discussion starts from introducing a function $u(x,y):\mathbb{R}^n\times\mathbb{R}^{1+a}\to\mathbb{R}$ is radially symmetric, i.e. for $|y|=|y'|=r$, we have $u(x,y)=u(x,y')$. I am working on ...
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### Solving the Radially Symmetric Poisson Equation with Exponential Source Term

I want to solve the poisson equation $$-\Delta u(\mathbf{x})=\rho(\mathbf{x})=\frac{e^{-|\mathbf{x}|^2}}{|\mathbf{x}|^2-1}$$ The problem want me to use the fundamental solution of laplace operator, ...
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### why the particular solution takes this form? [duplicate]

I am trying to find the particular solution of $$x^{\prime\prime}-3x^{\prime}-4x=5e^{-t}$$ I was always taught that the particular solution takes the same form as the non-homogenious part $5e^{-t}$. ...
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### Confusion on definition of fundamental solution for the heat equation

As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following ...
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### singular matrix and number of eigenvectors

what is the relationship between the a singular matrix and the number of linearly independent eigenvectors? i encountered this question in DE system, and here the number of linearly independent ...
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### Constant Coefficient Operators: Fundamental solution of Cauchy-Riemann Operator - Folland, Introduction to PDE

working on the exercises of section F. Constant-Coefficient Operators: Fundamental Solutions but I'm stuck in the following problem: I'm not sure how to use Green's theorem to conclude the second ...
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### Verification of the formula of variation of parameters for ODE

Background Consider the non homogeneous linear ODE in $\mathbb{R}^n$: $$\dot{\xi}(t) = A(t)\xi(t) + \nu(t)\; \xi(\sigma_0) = x_0 \label{1}\tag{I}$$ With $A$ a matrix of size $n\times n$ whose entries ...
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### Exact Solution form Fundamental Solution PDE

Let $f$ be a function on $\mathbb{R}_+\times\mathbb{R}^d$. Let $L$ be some differential operator, like $L=\frac{\partial}{\partial t}+\frac{\partial^2}{\partial x^2}$. Consider for some function $g$ ...
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We consider an differential equation $$d_A: \frac{dW}{dz} = A(z) W$$ with $A(z) \in \mathit{Mat}(n, \mathcal{O}_{\mathbb{C^*}})$ and $W(z)$ a fundamental system of this equation. If I now consider ...